Magnetic curves in the real special linear group

We investigate contact magnetic curves in the real special linear group of degree $2$. They are geodesics of the Hopf tubes over the projection curve. We prove that periodic contact magnetic curves in $\mathrm{SL}_2 \mathbb{R}$ can be quantized in the set of rational numbers. Finally, we study contact homogeneous magnetic trajectories in $\mathrm{SL}_2 \mathbb{R}$ and show that they project to horocycles in $\mathbb{H}^2 (-4)$.

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This work was partially supported by Kakenhi 15K04834 and 19K03461 (Japan).

The second-named author is supported by the project funded by the Ministry of Research and Innovation within Program 1 – Development of the national RD system, Subprogram 1.2 – Institutional Performance – RDI excellence funding projects, Contract no. 34PFE/19.10.2018.

Published 15 May 2020